calculus - Compute $int^1_0 int^{x^2}_0frac{y}{e^x} , dy , dx$

Compute $\displaystyle \int^1_0 \int^{x^2}_0 \frac{y}{e^x} \, dy \, dx$

It's been a couple of years since I've done any real integration and we just started doing double integrals in my Calculus 3 class. I can't remember what do do from here:

$$\int^1_0 \left.\frac{1}{e^x}\frac{y^2}{2}\right|^{x^2}_0 \, dx = \frac{1}{2} \int^1_0 \frac{x^4}{e^x} \, dx$$

From here I assumed integration by parts:

$$u=e^x$$

$$du=e^x \, dx$$

$$dv=4x^3 \, dx$$

$$v=x^4$$

Setting this up I get:

$$\frac{e^x}{2} x^4-\int^1_0 x^4e^x \, dx$$

This is where I'm stuck. I'm not sure where to go from here.

Answer

Let me suggest a different strategy. Take the integral
$$\int x^ne^{-x}\mathrm{d}x$$
Take
$$u=x^n,\quad\mathrm{d}u=nx^{n-1}\mathrm{d}x$$
$$\mathrm{d}v=e^{-x}\mathrm{d}x,\quad v=-e^{-x}$$
We then have
$$\int x^ne^{-x}\mathrm{d}x=vu-\int v\mathrm{d}u=-x^ne^{-x}+\int nx^{n-1}e^{-x}\mathrm{d}x$$
Notice how this reduces the power $x$ is raised to.

Can you apply this here and whittle down the power until the only term you have to integrate is

$$A\int e^{\pm x}\mathrm{d}x$$
for some constant $A$?

The reason I suggest this is that with your choice of $u$ and $\mathrm{d}v$, the power that $x$ is raised to will keep rising, which isn't helpful at all.